innovation and the financial guillotinefinancial guillotine 3 tolerance has a price. the most...

NBER WORKING PAPER SERIES

INNOVATION AND THE FINANCIAL GUILLOTINE

Ramana NandaMatthew Rhodes-Kropf

Working Paper 19379http://www.nber.org/papers/w19379

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138August 2013

We thank Gustavo Manso, Josh Lerner, Thomas Hellmann, Michael Ewens, Bill Kerr, Serguey Braguinsky,Antoinette Schoar, Bob Gibbons and Marcus Opp as well as seminar participants at CMU and MITfor fruitful discussion and comments. All errors are our own. The views expressed herein are thoseof the authors and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2013 by Ramana Nanda and Matthew Rhodes-Kropf. All rights reserved. Short sections of text,not to exceed two paragraphs, may be quoted without explicit permission provided that full credit,including © notice, is given to the source.

Innovation and the Financial GuillotineRamana Nanda and Matthew Rhodes-KropfNBER Working Paper No. 19379August 2013JEL No. G24,G39,O31

ABSTRACT

Our paper demonstrates that while failure tolerance by investors may encourage potential entrepreneursto innovate, financiers with investment strategies that tolerate early failure endogenously choose tofund less radical innovations. Failure tolerance as an equilibrium price that increases in the level ofexperimentation. More experimental projects that don't generate enough to pay the price cannot bestarted. In equilibrium all competing financiers may choose to offer failure tolerant contracts to attractentrepreneurs, leaving no capital to fund the most radical, experimental projects. The tradeoff betweenfailure tolerance and a sharp guillotine helps explain when and where radical innovation occurs.

Ramana NandaHarvard Business SchoolRock Center 317BostonMA [email protected]

Matthew Rhodes-KropfHarvard Business SchoolRock Center 313Soldiers FieldBoston, MA 02163and [email protected]

Innovation and the Financial Guillotine

Investors, corporations and even governments who fund innovation must decide which

projects to finance and when to withdraw their funding in order to create the most value.

A key insight from recent work is that a tolerance for failure may be extremely important

for innovation as it makes agents more willing to take risks and to undertake exploratory

projects that lead to innovation Holmstrom (1989), Aghion and Tirole (1994) and Manso

(2011). Agents penalized for early failure are less willing to experiment. Similarly Stein

(1989) argues that managers must be protected from short term financial reactions in

order to encourage them to make long-run investments.1

The optimal level of failure tolerance, of course, varies from project to project. Yet, in

many instances, a project-by-project optimization is not feasible. For example, a govern-

ment looking to stimulate innovation may pass laws making it harder to fire employees.

This level of ‘failure tolerance’ will apply to all employees, regardless of the projects they

are working on. Similarly, a CEO with a long-term, ‘failure tolerant’ employment contract

may take on many different types of projects. In fact, organizational structure, organi-

zational culture, or a desire by investors to build a consistent reputation as entrepreneur

friendly all result in firm-level policies towards failure tolerance. Put differently, the prin-

cipal often has an ‘innovation strategy’ that is set ex ante—one that is a blanket policy

that covers all projects in the principal’s portfolio—and hence may not be optimal for

every one of the projects. How does this financing strategy impact innovation? This is an

important question because the answer inverts how we think about the impact of failure

tolerance.

In this paper we depart from the prior literature that has looked at the optimal solu-

1A number of empirical papers consider the impact of policies that reduce managerial myopia and allow managersto focus on long-run innovation (Burkart, Gromb and Panunzi (1997), Myers (2000), Acharya and Subramanian(2009) , Ferreira, Manso and Silva (2011), Aghion, Reenen and Zingales (2009)).

2 JULY 2013

tion for individual projects, and instead consider the ex ante strategic choice of a firm,

investor or government looking to maximize profits or promote innovation. We examine

how different strategies impact the types of projects that an investor is willing to finance,

and how this may impact the nature of innovation that will be undertaken across different

types of firms and regions.

In particular, we highlight a central trade-off faced by principals when they pick their

innovation strategy. A strategy that is more failure tolerant may encourage the agent to

innovate, but simultaneously destroys the value of the real option to abandon the project.

In the real options literature (Gompers (1995), Bergemann and Hege (2005), Bergemann,

Hege and Peng (2008)), innovation is achieved through experimentation – several novel

ideas can be tried and only those that continue to produce positive information should

continue to receive funding. This idea has motivated the current thrust by several venture

capital investors to fund the creation of a “minimum viable product” in order to test new

entrepreneurial ideas as quickly and cheaply as possible, to ‘kill fast and cheap’, and only

commit greater resources to improve the product after seeing early success.2

Thus, a failure tolerant policy has two effects: it stimulates innovation which creates

value but destroys the value of the abandonment option. Put differently, a failure tolerant

strategy increases the entrepreneur’s willingness to experiment but decreases the investors

willingness to fund experimentation.

We show that financiers who are more tolerant of early failure endogenously choose to

fund less radical innovations, or ones where the value of abandonment options is low. This

is because although entrepreneurs prefer a failure tolerant investor, in equilibrium, failure

2Venture capital investors seem to have sharp ready guillotines - Sahlman (1990), Hellmann (1998); Gompersand Lerner (2004) document the myriad control rights negotiated in standard venture capital contracts that allowinvestors to fire management and/or abandon the project. In fact, Hall and Woodward (2010) document that about50% of the venture-capital backed startups in their sample had zero-value exits. Hellmann and Puri (2002) andKaplan, Sensoy and Stromberg (2009) show that of the firms that are ‘successful’, many end up with CEOs whoare different from the founders.

FINANCIAL GUILLOTINE 3

tolerance has a price. The most radical projects cannot afford to pay the price. Thus,

the most radical innovations are either not funded at all, or are endogenously funded by

financiers who have a sharp financial guillotine.3 In fact, we show that principals have

to be careful, since a strategy of being failure tolerant to promote innovation may have

exactly the opposite effect than the one desired, leading to the funding of less radical

innovation.

We also demonstrate that an equilibrium can arise in which all competing financiers

choose to be failure tolerant in the attempt to attract entrepreneurs and thus no capital

is available to fund the most radical innovations, even if there are entrepreneurs who want

to find financing for such projects. This equilibrium becomes more likely to form when

entrepreneurs on average have a greater desire for failure tolerance such as is thought to

occur, for example, in parts of Europe and Japan (see Landier (2002)4). Moreover, the

equilibrium with all failure tolerant investors may be self-fulfilling if the act of shutting

down more projects reduces the stigma attached to failure.

Our model therefore highlights that the type of innovation undertaken in an economy

may depend critically on the institutions that either facilitate or hinder the ability to

terminate projects at an intermediate stage, as well as cultural or institutional factors that

determine the outside options for entrepreneurs. The institutional funding environment

for innovation is an endogenous equilibrium outcome that may result in places or times

with no investors able to fund radical innovation.

This paper is related to prior work examining the role of principal agent relationships in

the innovation process (e.g. Holmstrom (1989), Aghion and Tirole (1994), Hellmann and

Thiele (2011) and Manso (2011)) as well as how the principle agent problem influences

3Our model also demonstrates that some radical innovations can only be commercialized by investors who arenot concerned with making NPV positive investments, such as for example, government funded initiates like themanhattan project or the lunar landing.

4In Landier (2002) the stigma of failure prevents entrepreneurs from abandoning bad projects.

4 JULY 2013

the decision to stop funding projects (e.g. Bergemann and Hege (2005), Cornelli and

Yosha (2003) and Hellmann (1998)). We build on this work by considering the type of

project an investor is willing to fund given their strategy (due to ability or willingness) to

end the project at an intermediate stage. Our work is also related to research examining

how incentives stemming from organizational structure can drive innovation (e.g. Qian

and Xu (1998), Gromb and Scharfstein (2002), Fulghieri and Sevilir (2009)) and how

the “soft budget constraint” problem drives the selection of projects (e.g. Roberts and

Weitzman (1981) and Dewatripont and Maskin (1995)). We look specifically at innovation

as an outcome and examine how these factors impact the degree to which investors choose

to fund radical innovation. Finally, a recent group of empirical papers have looked for

and found a positive effect of failure tolerance on the margin (e.g. Lerner and Wulf

(2007), Azoulay, Zivin and Manso (2011), Acharya and Subramanian (2009), Ferreira,

Manso and Silva (2011), Aghion, Reenen and Zingales (2009), Tian and Wang (2012),

Chemmanur, Loutskina and Tian (2012)). Our ideas are consistent with these findings,

although different from past theoretical work, as our point is that strategies that reduce

short term accountability and thus encourage innovation on the intensive margin may

simultaneously alter what financial backers are willing to fund and thus reduce innovation

at the extensive margin. Examining this latter effect seems to be a fruitful avenue for

further empirical research.5

The tradeoff we explore also has implications for a wider array of situations than just

R&D. In the context of a board choosing a CEO, the intuition presented here suggests

that boards that provide long term contracts with more tolerance for failure may find that

they then choose a more experienced CEO who is a more known commodity. A board

5Recent work by Ewens and Fons-Rosen (2013) and Cerqueiro et al. (2013) have found initial support for thethe idea that failure tolerance may encourage innovation at the intensive margin but discourage it at the extensivemargin.


that makes it easy to fire the CEO is more likely to experiment by hiring a younger, less

experienced CEO whose quality is less certain but whose potential may be great. Thus, the

same result occurs in this context - the desire to alter the intensive margin for innovation

alters the extensive margin in the willingness to select a more radical leader.

The remainder of the paper is organized as follows. Section I develops a model of

investing in innovative projects from both the financier’s and entrepreneur’s point of view.

Section II solves for the deal between the financier and entrepreneur for different types of

projects and levels of commitment. Section III determines the choices of the entrepreneur

and investor given their level of commitment and desire for a committed investor. Section

IV endogenizes the choice of failure tolerance by the investor and determines the potential

equilibria and how they depend on the the view of early failure in the labor market and

by the entrepreneur. Section V discusses the key implications and extensions of our model

and Section VI concludes.

I. A Model of Investment

We model the creation of new projects that need an investor and an entrepreneur in

each of two periods. Both the investor and entrepreneur must choose whether or not to

start a project and then, at an interim point, whether to continue the project.

This basic set up is a two-armed bandit problem. There has been a great deal of

work modeling innovation that has used some from of the two armed bandit problem.

From the classic works of Weitzman (1979), Roberts and Weitzman (1981), Jensen (1981),

Battacharya, Chatterjee and Samuelson (1986) to more recent works such as Moscarini

and Smith (2001), Manso (2011) and Akcigit and Liu (2011).6 We build on this work by

altering features of the problem to explore an important dimension in the decision to fund

6See Bergemann and Valimaki (2006) for a review of the economics literature on bandit problems.

6 JULY 2013

innovation.

A. Investor View

We model investment under uncertainty. A penniless entrepreneur seeks funding from

investors for a risky project that requires capital for two periods or stages. The first stage

of the project reveals information about the probability of success in the second stage.7

The probability of ‘success’ (positive information) in the first stage is p1 and reveals the

information S, while ‘failure’ reveals F . Success in the second stage yields a payoff of VS

or VF depending on what happened in the first stage, but occurs with a probability that

is unknown and whose expectation depends on the information revealed by the first stage.

Failure in the second stage yields a payoff of zero.

Let E[p2] denote the unconditional expectation about the second stage success. The

investor updates their expectation about the second stage probability depending on the

outcome of the first stage. Let E[p2|S] denote the expectation of p2 conditional on success

in the first stage, while E[p2|F ] denotes the expectation of p2 conditional on failure in the

first stage.8

The project requires capital X to complete the first stage of the project and Y to

complete the second stage. The entrepreneur is assumed to have no capital while the

investor has enough to fund the project for both periods. An investor who chooses not

to invest at either stage can instead earn a safe return of r per period (investor outside

option) on either X, Y or both. We assume project opportunities are time sensitive, so if

the project is not funded at either the 1st or 2nd stage then it is worth nothing.

7This might be the building of a prototype or the FDA regulated Phase I trials on the path of a new drug. Etc.8One particular functional form that is sometimes used with this set up is to assume that the first and second

stage have the same underlying probability of success, p. In this case p1 can be thought of as the unconditionalexpectation of p, and E[p2|S] and E[p2|F ] just follow Bayes’ rule. We use a more general setup to express the ideathat the probability of success of the first stage experiment is potentially independent of the amount of informationrevealed by the experiment. For example, there could be a project for which a first stage experiment would workwith a 20% chance but if it works the second stage is almost certain to work (99% probability of success).


In order to focus on the interesting cases we assume that if the project ‘fails’ in the first

period then it is NPV negative in the second period, i.e., E[p2|F ] ∗ VF < Y (1 + r). And

if the project ‘succeeds’ in the first period then it is NPV positive in the second period,

i.e., E[p2|S] ∗VS > Y (1 + r). We will consider how variation in the probabilities alters the

decision to fund the project.

We assume that principals/investors cannot commit to a long-term contract with the

agent/entrepreneur at the beginning of the project. Prior work has assumed an idealized

investor who can write long-term contracts allowing them to commit to some projects

and not to others. Our departure from this work allows us to compare investors who

can never commit to committed investors introduced in the next section. With limited

commitment, the principal and the agent may agree on and bind themselves to short-term

(one period) contracts, but cannot commit themselves to any future contracts. Investors

in new projects are often unable to commit to fund the project in the future even if they

desire to make such a commitment. For example, corporations cannot write contracts

with themselves and thus always retain the right to terminate a project. Venture capital

investors have strong control provisions for many standard incomplete contracting reasons

and are unable to give up the power to shut down the firm and return any remaining

money if they wish to do so in the future. Thus, even a project that receives full funding

(both X and Y) in the first period, may be shut down and Y returned to investors in

period two.

We will demonstrate that the equilibrium fraction owned by the investor in the final

period, assuming an agreement can be reached for investment in both periods, depends on

the outcome of the first period. Let αS represent the final fraction owned by the investors

if the first period was a success, and let αF represent the final fraction owned by the

investors if the first period was a failure.

8 JULY 2013

The extensive form of the game played by the investor (assuming the entrepreneur is

willing to start and continue the project) is shown in figure 1. Remember that by choosing

not to invest in the project in either period the investor earns a return of r per period on

the money he does not invest in the risky project. We assume investors make all decisions

to maximize net present value (which is equivalent to maximizing end of second period

wealth).

P1

1 – p1

Invest $X?

Failure, F

Invest $Y?

1 – E[p2 | S]

E[p2 | S]

Failure, Payoff

0

Success, Payoff

VS*αS Success, S

Invest $Y?

1 – E[p2 |F]

E[p2 | F]

Failure, Payoff

0

Success, Payoff

VF*αF

Yes

No

X (1+r)2 + Y (1+r)

Yes

No

No

Y (1+r)

Y (1+r)

Yes

Figure 1. Extensive Form Representation of the Investor’s Game Tree

B. Entrepreneur’s View

Potential entrepreneurs are endowed with a project in period one with a given p1, p2,

E[p2|S], E[p2|F ], VS , VF , X and Y . Assuming that an investor chooses to fund the first

period of required investment, the potential entrepreneur must choose whether or not to

become an entrepreneur or take an outside employment option. If the investor is willing to

fund the project in the second period then the entrepreneur must choose whether or not to

continue as an entrepreneur. If the potential entrepreneur chooses entrepreneurship and

stays an entrepreneur in period 2 they generate utility of uE in both periods. Alternatively,

if they choose not to become an entrepreneur in the first period then we assume that no

entrepreneurial opportunity arises in the second period so they generate utility of uO in


both periods.9

If the investor chooses not to fund the project in the second period, or the entrepreneur

chooses not to continue as an entrepreneur, i.e., the entrepreneur cannot reach an agree-

ment with an investor in period 2, then the project fails and the entrepreneur generates

utility uF from their outside option in the second period. We assume ∆uF = uF −uE < 0,

which represents the disutility felt by a failed entrepreneur. The more negative ∆uF is,

the worse entrepreneurial experience in a failed project is perceived.10

Given success or failure in the first period the entrepreneur updates their expectation

about the probability the project is a success just as the investor does. The extensive form

of the game played by the entrepreneur (assuming funding is available) is shown in figure

2. We assume entrepreneurs make all decisions to maximize the sum of total utility.

P1

1 – p1

Start Firm?

Failure, F

Continue?

1 – E[p2 | S]

E[p2 | S]

Failure, Payoff 2 uE

Success, Payoff

VS*(1-αS) +2 uE

Success, S

Continue?

1 – E[p2 |F]

E[p2 | F]

Failure, Payoff 2 uE

Success, Payoff

VF*(1-αF) + 2 uE

Yes

No

2uO

Yes

No

No

uE + uF

uE + uF

Yes

Figure 2. Extensive Form Representation of the Entrepreneur’s Game Tree

9The entrepreneur could also receive side payments from the investor. This changes no results and so is sup-pressed.

10Entrepreneurs seem to have a strong preference for continuation regardless of present-value considerations, beit because they are (over)confident or because they rationally try to prolong the search. Cornelli and Yosha (2003)suggest that entrepreneurs use their discretion to (mis)represent the progress that has been made in order to securefurther funding.

10 JULY 2013

II. The Deal Between the Entrepreneur and Investor

In this section we determine when the entrepreneur and investors will be able to find an

acceptable deal. We do so by determining the minimum share both the entrepreneur and

investor must own in order to choose to start the project.

The final fraction owned by investors after success or failure in the first period, αj where

j ∈ S, F, is determined by the amount the investors purchased in the first period, α1,

and the second period α2j , which may depend on the outcome in the first stage. Since the

first period fraction gets diluted by the second period investment, αj = α2j +α1(1−α2j).

A. No Commitment

Using backward induction we start with the second period. Conditional on a given α1

the investor will invest in the second period as long as

VjαjE[p2 | j]− Y (1 + r) > 0 where j ∈ S, F

This condition does not hold after failure even if αF = 1, therefore the investor will only

invest after success in the first period. The minimum fraction the investor is willing to

accept for an investment of Y in the second period after success in the first period is

α2S =Y (1 + r)

VSE[p2 | S].

The entrepreneur, on the other hand, will continue with the business in the second

period as long as,

Vj(1− αj)E[p2 | j] + uE > uF where j ∈ S, F.


The entrepreneur will want to continue if the expected value from continuing is greater

than the utility after failure, because the utility after failure is the outside option of the

entrepreneur if she does not continue. The maximum fraction the entrepreneur will give

up in the second period after success in the first period is

α2S = 1− uF − uEVSE[p2 | S]

.

Given both the minimum fraction the investor will accept, α2S , as well as the maximum

fraction the entrepreneur will give up, α2S , an agreement may not be reached. An investor

and entrepreneur are able to reach an agreement in the second period as long as

1 ≥ α2S ≤ α2S ≥ 0 Agreement Conditions, 2ndperiod

The middle inequality requirement is that there are gains from trade. However, those

gains must also occur in a region that is feasible, i.e. the investor requires less than 100%

ownership to be willing to invest, 1 ≥ α2S , and the entrepreneur requires less than 100%

ownership to be willing to continue, α2S ≥ 0. If not, the entrepreneur, for example, might

be willing to give up 110% of the final payoff and the investor might be willing to invest

to get this payoff, but it is clearly not economically feasible. For the same reason, even

when there are gains from trade in the reasonable range, the resulting negotiation must

yield a fraction such that 0 ≤ α2j ≤ 1 otherwise it is bounded by 0 or 1.

We could find the maximum fraction the entrepreneur would be willing to give up after

failure (α2F ), however, we already determined that the investor would require a share

(α2F ) greater than 100% to invest in the second period, which is not economically viable.

So no deal will be done after failure.

12 JULY 2013

If an agreement cannot be reached even after success then clearly the deal will never

be funded. However, even those projects for which an agreement could be reached after

success may not be funded in the first period if the probability of success in the first period

is too low. The following proposition determines the conditions for a potential agreement

to be reached to fund the project in the first period. Given that the investor can forecast

the second period dilution these conditions can be written in terms of the final fraction of

the business the investor or entrepreneur needs to own in the successful state in order to

be willing to start.

PROPOSITION 1: The minimum total fraction the investor must receive is

αSN=p1Y (1 + r) +X(1 + r)2

p1VSE[p2 | S]

and the maximum total fraction the entrepreneur is willing to give up is

αSN= 1− (1 + p1)(uO − uE) + (1− p1)(uO − uF )

p1VSE[p2 | S]

where the N subscript represents the fact that no agreement will be reached after failure.

See appendix A.ii for proof. We use the N subscript because in the next section we consider

the situation when reputation concerns or bureaucracy result in an agreement to continue

even after first period failure (A subscript for Agreement rather than N for No-agreement).

Then we will compare the deals funded in each case. Given the second period fractions

found above, the minimum and maximum total fractions imply minimum and maximum

first period fractions (found in the appendix for the interested reader).


B. Commitment

With the assumption of incomplete contracts there is potential value to an investor of

a reputation as ‘entrepreneurial friendly’ or ‘committed’, who might then find it costly

to shut down a project in the second period. Or alternatively, there might be value in

a bureaucratic institution that has a limited ability to shut down a project once started.

In this subsection we examine committed investors with an assumed (reputation) cost of

early shutdown of c. Then in section IV we allow investors to choose whether or not to

have a committed reputation.

We define a committed investor as follows.

DEFINITION 1: A committed investor has a c > c∗ = Y (1 + r)− VFE[p2 | F ]

That is, the cost of shutdown is greater than the expected loses from the project after

failure in the first period.

The following proposition solves for the minimum fraction the committed investor will

accept in the second period and the maximum fraction the entrepreneur will give up in

the second period. These will be used to determine if a deal can be reached.

PROPOSITION 2: The minimum fraction the committed investor is willing to accept for

an investment of Y in the second period after success in the first period is

α2S =Y (1 + r)

VSE[p2 | S].

However, after failure in the first period the minimum fraction the committed investor is

willing to accept is

α2F =Y (1 + r)− c

VFE[p2 | F ](1− α1)− α1

1− α1.

14 JULY 2013

The maximum fraction the entrepreneur will give up in the second period after success

in the first period is


.

After failure in the first period the maximum fraction the entrepreneur is willing to give

up is

α2F = 1− uF − uEVFE[p2 | F ](1− α1)

.

The proof is in appendix A.i. Both the investor and the entrepreneur must keep a large

enough fraction in the second period to be willing to do a deal rather than choose their

outside option. These fractions of course depend on whether or not the first period exper-

iment worked.

After success in the first period the agreement conditions are always met. However, after

failure in the first period the agreement conditions may or may not be met depending on

the parameters of the investment, the investor and the entrepreneur.

LEMMA 1: An agreement can be reached in the second period after failure in the first iff

the investor is committed.

PROOF:

A second period deal after failure can be reached if α2F − α2F ≥ 0.

α2F − α2F = 1− uF − uEVFE[p2 | F ](1− α1)

− Y (1 + r)− cVFE[p2 | F ](1− α1)

− α1

1− α1.

α2F − α2F is positive iff VFE[p2 | F ] − uF + uE − Y (1 + r) + c ≥ 0. However, since

the utility of the entrepreneur cannot be transferred to the investor, it must also be the

case that VFE[p2 | F ] − Y (1 + r) + c ≥ 0. But if VFE[p2 | F ] − Y (1 + r) + c ≥ 0 then

VFE[p2 | F ]− uF + uE − Y (1 + r) + c ≥ 0 because uF − uE < 0. QED


This lemma makes it clear that only a ‘committed’ investor will continue to fund the

company after failure because VFE[p2 | F ]− Y (1 + r) < 0.11

We have now solved for both the minimum second period fraction the committed investor

will accept, α2j , as well as the maximum second period fraction the entrepreneur will give

up, α2j , and the conditions under which a second period deal will be done. If either party

yields more than these fractions, then they would be better off accepting their outside,

low-risk, opportunity rather than continuing the project in the second period.

Stepping back to the first period, a committed investor will invest and an entrepreneur

will start the project with a committed investor only if they expect to end up with a large

enough fraction after both first and second period negotiations.

PROPOSITION 3: The minimum total fraction the investor is willing to accept is

αSA=Y (1 + r) +X(1 + r)2 − (1− p1)VFαFE[p2 | F ]

p1VSE[p2 | S],

and the maximum fraction the entrepreneur is willing to give up is

αSA= 1− 2∆w1 − (1− p1)E[p2 | F ]VF (1− αF )

p1VSE[p2 | S]

where the subscript A signifies that an agreement will be reached after first period failure.

And where

αF = γ

[Y (1 + r)− cVFE[p2 | F ]

]+ (1− γ)

[1− ∆uF

VFE[p2 | F ]

]The proof is in A.ii, however, these are the relatively intuitive outcomes in each situation

because each player must expect to make in the good state an amount that at least equals

their expected cost plus their expected loss in the bad state.

11Furthermore, at c = Y (1 + r), the committed investor will continue to fund after failure since VFE[p2 | F ] > 0.Thus, there is some c such that the investor is committed.

16 JULY 2013

Given the minimum and maximum fractions, we know the project will be started if

1 ≥ αSi ≤ αSi ≥ 0 Agreement Conditions, 1st period,

either with our without a second period agreement after failure (i ∈ [A,N ]).

We have now calculated the minimum and maximum required by investors and entre-

preneurs. With these fractions we can determine what kinds of deals will be done by the

different types of player.

III. The Desires of the Entrepreneur and Investor

It is informative to start by considering only the desires of the entrepreneur. The

entrepreneur is deciding whether to start the company or take the safe outside option.

We have calculated above the fraction of equity the entrepreneur will give up with and

without commitment from the investor, αSAand αSN

. Our next simple proposition uses

these to determine when an entrepreneur would want a committed or failure tolerant

investor. Remember that above we defined a ‘failure tolerant’ or ‘committed’ investor

as one who would still be willing to invest after failure in the first period, i.e., VFE[p2 |

F ]− Y (1 + r) + c ≥ 0.

PROPOSITION 4: The entrepreneur is willing to give up a larger fraction of the new

venture with a committed investor.

PROOF:

The entrepreneur is willing to give a larger fraction of the new venture to a failure

tolerant investor if

αSA− αSN

=(1− p1)γ [VFE[p2 | F ]− Y (1 + r) + c−∆uF ]

p1VSE[p2 | S]> 0.


The investor is only failure tolerant, i.e., willing to invest after failure if VFE[p2 | F ] −

Y (1 + r) + c ≥ 0. Given that ∆uF < 0, i.e. the entrepreneur finds early failure painful,

αSA−αSN

is positive if the investor is failure tolerant. Furthermore, lemma 1 implies that

a deal will always be done after early failure when this is true. QED

uF represents the cost of early failure to the entrepreneur. It is intuitive that if early

failure is costly to the entrepreneur then they prefer a failure tolerant investor and are

more willing to start a new innovative venture with a failure tolerant investor.

This proposition supports the intuition behind failure tolerance. Greater failure tol-

erance by the investor increases the willingness of the entrepreneur to choose the risky,

innovative path. This idea is correct but as the following proposition shows, there is a

force coming from the investor that works against this effect.

PROPOSITION 5: The investor is willing to accept a smaller fraction of the new venture

if the investor is uncommitted.

For the proof see Appendix A.iv.

Both proposition 4 and 5 are partial equilibrium results that demonstrate common

intuition about the two sides of the innovation problem when we consider them separately.

The entrepreneur is more willing to start an innovative project for a given offer from the

investor if the project will not be shut down after early failure. In this sense failure

tolerance encourages innovation.

At the same time proposition 5 demonstrates that the investor is more willing to fund

the project if they retain the option to shut down the project after early failure, i.e. real

options have value. But this elucidates the clear tension - the investor is more likely to

fund the project if he can kill it but the entrepreneur is more likely to start the project if

it wont get killed.

18 JULY 2013

To understand the interaction we must solve for the general equilibrium considering

both the entrepreneur and the investor.

A. Commitment or the Guillotine

A deal can be done to begin the project if αSA≤ αSA

, assuming an agreement will

be reached to continue the project after early failure. Alternatively, a deal can be done

to begin the project if αSN≤ αSN

, assuming the project will be shut down after early

failure. That is, a deal can get done if the lowest fraction the investor will accept, αSi is

less than the highest fraction the entrepreneur with give up, αSi . Therefore, given that a

second period agreement after failure will or will not be reached, a project can be started

if αSA− αSA

≥ 0, i.e., if

p1VSE[p2 | S] + (1− p1)VFE[p2 | F ]− 2(uO − uE)− Y (1 + r)−X(1 + r)2 ≥ 0, (1)

or if αSN− αSN

≥ 0, i.e., if

p1VSE[p2 | S]− 2(uO − uE) + (1− p1)∆uF − p1Y (1 + r)−X(1 + r)2 ≥ 0. (2)

We can use the above inequalities to determine what types of projects actually can be

started and the effects of failure tolerance and a sharp guillotine.

PROPOSITION 6: For any given project there are four possibilities

1) the project can only be started if the investor is committed,

2) the project can only be started if the investor has a sharp guillotine (is uncommitted),

3) the project can be started with either a committed or uncommitted investor,


4) the project cannot be started.

The proof is left to Appendix A.v. Proposition 6 demonstrates the potential for a tradeoff

between failure tolerance and the launching of a new venture. While the entrepreneur

would like a committed investor the commitment comes at a price. For some projects and

entrepreneurs that price is so high that they would rather not do the deal. For others they

would rather do the deal, but just not with a committed investor. Thus when we include

the equilibrium cost of failure tolerance we see that it has the potential to both increase

the probability that an entrepreneur chooses the innovative path and decrease it.

Essentially the utility of the entrepreneur can be enhanced by moving some of the payout

in the success state to the early failure state. This is accomplished by giving a more failure

tolerant VC a larger initial fraction in exchange for the commitment to fund the project

in the bad state. If the entrepreneur is willing to pay enough in the good state to the

investor to make that trade worth it to the investor then the deal can be done. However,

there are deals for which this is true and deals for which this is not true. If the committed

investor requires too much in order to be failure tolerant in the bad state, then the deal

may be more likely to be done by a VC with a sharp guillotine.

B. Who Funds Experimentation?

The question is then – which projects are more likely to be done by a committed or

uncommitted investor? We can see that projects with higher payoffs, VS or VF , or lower

costs, Y and X, are more likely to be done, but when considering the difference between a

committed and an uncommitted investor we must look at the value of the early experiment.

In our model the first stage is an experiment that provides information about the prob-

ability of success in the second stage. In an extreme one might have an experiment that

demonstrated nothing, i.e., VSE[p2 | S] = VFE[p2 | F ]. That is, whether the first stage

20 JULY 2013

experiment succeeded or failed the updated expected value in the second stage was the

same. Alternatively, the experiment might provide a great deal of information. In this

case VSE[p2 | S] would be much larger than VFE[p2 | F ]. Thus, VSE[p2 | S]−VFE[p2 | F ]

is the amount or quality of the information revealed by the experiment.

We define a project as more experimental if the first stage reveals more information.

This definition is logical since VSE[p2 | S]−VFE[p2 | F ] is larger if the experiment revealed

more about what might happen in the future. In the one extreme the experiment revealed

nothing so VSE[p2 | S] − VFE[p2 | F ] = 0. At the other extreme the experiment could

reveal whether or not the project is worthless (VSE[p2 | S]− VFE[p2 | F ] = VSE[p2 | S]).

One special case are martingale beliefs with prior expected probability p for both stage

1 and stage 2 and E[p2 | j] follows Bayes Rule. In this case projects with weaker priors

would be classified as more experimental.

While this is a logical definition of increased experimentation, increasing VSE[p2 | S]−

VFE[p2 | F ] might simultaneously increase or decrease the total expected value of the

project. When we look at the effects from greater experimentation we want to make sure

that we hold constant any change in expected value. Therefore, we define a project as

more experimental in a mean preserving way as follows.

DEFINITION 2: A project is more experimental in a mean preserving way if VSE[p2 |

S] − VFE[p2 | F ] is larger for a given p1, and expected payoff, p1VSE[p2 | S] + (1 −

p1)VFE[p2 | F ].

We use this definition because it changes the level of experimentation without simulta-

neously altering the probability of first stage success or the expected value of the project.

Certainly a project may be more experimental if VSE[p2 | S] − VFE[p2 | F ] is larger and

the expected value is larger.12 However, this kind of difference would create two effects -

12For example, if E[p2 | F ] is always zero, then the only way to increase VSE[p2 | S]− VFE[p2 | F ] is to increase


one that came from greater experimentation and one that came from increased expected

value. Since we know the effects of increased expected value (everyone is more likely to

fund a better project) we use a definition that isolates the effect of information.

Note that the notion of increasing experimentation has a relation to, but is not the same

as, increasing risk. For example, we could increase risk while holding the experimentation

constant by decreasing both E[p2 | S] and E[p2 | F ] and increasing VS and VF . This

increase in risk would increase the overall risk of the project but would not impact the

importance of the first stage experiment.

With this definition we can establish the following proposition

PROPOSITION 7: A more experimental project is more likely to be funded by an un-

committed investor. A more experimental project can potentially only be funded by an

uncommitted investor.

PROOF:

See Appendix A.vi

Proposition 7 makes it clear that the more valuable the information learned from the

experiment the more important it is to be able to act on it. A committed investor cannot

act on the information and must fund the project anyway while an uncommitted investor

can use the information to terminate the project. Therefore, an increase in failure tolerance

decreases an investors willingness to fund projects with greater experimentation.

Figure 3 demonstrates the ideas in propositions 6 and 7. Projects with a given expected

payoff after success in the first period (Y-axis) or failure in the first period (X-axis) fall into

different regions or groups. We only examine projects above the 45 line because it is not

economically reasonable for the expected value after failure to be greater than the expected

VSE[p2 | S]. In this case the project will have a higher expected value and be more experimental. We are not rulingthis possibilities out, rather we are just isolating the effect of experimentation.

22 JULY 2013

value after success. In the upper left diagram the small dashed lines that run parallel to

E[p2 |F] VF

E[p2 |S] V

S

45°

K

C

N

A’

A

E[p2 |F] VF

E[p2 |S] V

S

45°

K

N

A’

A

E[p2 |F] VF

E[p2 |S] V

S

45°

C

N

A’

A

E[p2 |F] VF

p1 = 0.4

E[p2 |S] V

S

45°

33°

Iso Expected Payoff lines

Figure 3. Investor regions: N = No Investors, C = Only Committed Investors, K = Only Killer In-

vestors, A = All Investors, A’ = All Invest, Neither Kills

the 45 line are Iso Experimentation lines, i.e., along these lines VSE[p2 | S]−VFE[p2 | F ]

is constant. These projects can be thought of as equally experimental. Moving northeast

along an Iso Experimentation line increases the project’s value without changing the degree

to which it is experimental.

The large dashed lines are Iso Expected Payoff lines. These projects have the same ex

ante expected payoff, p1VSE[p2 | S]+(1−p1)VFE[p2 | F ]. They have a negative slope that

is defined by the probability of success in the first period p1.13 Projects to the northwest

13In the example shown p1 = 0.4 so the slope of the Iso Expected Payoff Lines is −1.5 resulting in a angle to theY-axis of approximately 33 degrees.


along an Iso Expected Payoff line are more experimental, but have the same expected

value.

The diagram also reinforces that risk is distinct from our notion of experimentation.

Each point in the diagram could represent a more or less risky project. A project with a

higher VS and VF but lower E[p2 | S] and E[p2 | F ] would be much more risky but could

have the same VSE[p2 | S] and VFE[p2 | F ] as an alternative less risky project. Thus,

these two different projects would be on the same Iso Experimentation and Iso Expected

Payoff line with very different risk.

In the remaining three diagrams in Figure 3 we see the regions discussed in proposition

6. The large dashed line is defined by equation (1). Above this line αSA−αSA

≥ 0, so the

entrepreneur can reach an agreement with a committed investor. Committed investors

will not invest in projects below the large dashed line and can invest in all projects above

this line. This line has the same slope as an Iso Expected Payoff Line because with

commitment the project generates the full ex ante expected value. However, with an

uncommitted, or killer, investor the project is stopped after failure in the first period.

Thus, the killer investor’s expected payoff is independent of VFE[p2 | F ]. Therefore,

uncommitted investors will invest in all projects above the horizontal dotted line. This

line is defined by equation (2) because a killer and an entrepreneur can reach an agreement

as long as αSN− αSN

≥ 0. The vertical line with both a dot and dash is the line where

VFE[p2 | F ] = Y (1 + r). Projects to the right of this line (region A’) have a high enough

expected value after failure in the first period that no investor would ever kill the project,

so we focus our attention to the left of this line.14

Where, or whether the dotted and dashed lines cross depends on the other parameters

in the problem (c, uO, uE , uF , r,X, Y ) that are held constant in each diagram. If the lines

14We have assumed throughout the paper that VFE[p2 | F ] < Y (1 + r) to focus on the interesting cases wherekilling and commitment matter.

24 JULY 2013

cross, as in the upper right diagram, we see five regions. Entrepreneurs with projects with

high enough expected values can reach agreements with either type of investor (region

A) and those with low enough expected values cannot find investors (region N). However,

projects with mid level expected values may only be able to reach an agreement with

one of the two types of investors. This displays the intuition of proposition 7. We see

that projects with a given level of expected payoff are more likely to be funded only by

a killer (region K) if they are more experimental and more likely to be funded only by a

committed investor (region C) if they are less experimental.

The following corollary follows directly from proposition 7 and the intuition in the

diagrams. If the entrepreneur has a greater dislike of early failure, uF is lower, then the

dotted line is higher and the C region expands and the K region shrinks. In the lower left

and right diagrams in Figure 3 we see it is possible, depending on the magnitude of uF

for region C or K to disappear. Formally,

COROLLARY 1: Entrepreneurs with a greater dislike of early failure, (smaller uF ), are

more likely to have a project such that they can only reach an agreement with a committed

investor.

This key result and corollary seem contrary to the notion that failure tolerance increases

innovation (Holmstrom (1989), Aghion and Tirole (1994) and Manso (2011)), but actually

fits both with this intuition and with the many real world examples. The source of many of

the great innovations of our time come both from academia or government labs, places with

great failure tolerance but with no criteria for NPV-positive innovation, and from venture

capitalist investments, a group that cares a lot about the NPV of their investments, but is

often reviled by entrepreneurs for their quickness to shut down a firm. On the other hand,

many have argued that large corporations, that also need to worry about the NPV of their


investments, engage in more incremental innovation and are slow to kill projects.15

Our model helps explain this by highlighting that having a strategy of a sharp guillotine

allows investors to back the most experimental projects, or those associated with the most

radical innovation. Proposition 7 tells us that corporate investors, whose bureaucracy may

make them slow to kill projects, will tend to fund projects that are ex ante less experimental

(and so wont need to kill them). While VCs, who are generally faster with the financial

guillotine, will, on average, fund things with greater learning from early experiments and

kill those that don’t work out.16 Thus, even though corporations will have encouraged

more innovation they will only have funded the less experimental projects. And the VCs

will have discouraged entrepreneurs from starting projects ex ante. However, ex post they

will have funded the most experimental projects and thus will produce the more radical

innovations! On the other hand, failure tolerance can induce entrepreneurs to engage in

experimentation, but the price of being a failure tolerant investor who cares about NPV

may be too high - so that institutions such as academia and the government may also be

places that end up financing a lot of radical experimentation, just not in an NPV positive

way.17

Our model also suggests that employees will likely complain about the stifling environ-

ment of the corporation that does not let them innovate (because the corporation won’t

fund very experimental projects) – leading to spinoffs due to frustration and disagreements

about what projects to push forward (see Gompers, Lerner and Scharfstein (2005) and

15For example, systematic studies of R&D practices in the U.S. report that large companies tend to focus R&Don less uncertain, less novel projects more likely to focus on cost reductions or product improvement than newproduct ideas (e.g. Henderson (1993), Henderson and Clark (1990), Scherer (1991), Scherer (1992), Jewkes, Sawersand Stillerman (1969) and Nelson, Peck and Kalachek (1967)).

16Hall and Woodward (2010) report that about 50% of the venture-capital backed startups in their sample hadzero-value exits

17Recent work, Chemmanur, Loutskina and Tian (2012), has reported that corporate venture capitalists seem tobe more failure tolerant than regular venture capitals. Interestingly, corporate venture capitalists do not seem tohave had adequate financial performance but Dushnitsky and Lenox (2006) has shown that corporations benefit innon-pecuniary ways (see theory by Fulghieri and Sevilir (2009)). Our theory suggests that as the need for financialreturn diminishes, investors can become more failure tolerant and promote innovation.

26 JULY 2013

Klepper and Sleeper (2005)). Corporations that want to retain and fund more radical

innovations likely need to become less failure tolerant.18 In so doing they will become

more willing to fund very experimental projects.

Remember that the notion of increasing experimentation is not the same as increasing

risk. Thus, our point is not that more failure tolerant investors, such as corporations, will

not do risky projects. Rather they will be less likely to take on projects with a great deal

of experimentation and incremental steps where in a great deal of the project value comes

from the ability to kill it.

IV. Investors choice of commitment level

In this section we endogenize the choice by the investor to become committed or not.

This allows us to demonstrate the potential for different investing equilibrium environ-

ments.

A. The Search for Investments and Investors

We model the process of the match between investors and entrepreneurs using a sim-

plified version of the classic search model of Diamond-Mortensen-Pissarides (for examples

see Diamond (1993) and Mortensen and Pissarides (1994) and for a review see Petrongolo

and Pissarides (2001)).19 This allows the profits of the investors to vary depending on

how many others have chosen to be committed or quick with the guillotine.

We assume that there are a measure of of investors, MI , who must choose between

having a sharp guillotine, c = 0, (type K for ‘killer’) or committing to fund the next

round (type C for committed). Simultaneously we assume that there are a measure of

18Interestingly, Seru (2011) reports that mergers reduce innovation. This may be because the larger the corpora-tion the more failure tolerant it becomes and thus endogenously the less willing it becomes to fund innovation.

19For a complete development of the model see Pissarides (1990). A search and Nash bargaining combinationwas recently used by Inderst and Muller (2004) in examining venture investing.


entrepreneurs, Me, with one of two types of projects, type A and B. Type A projects

occur with probability φ, while the type B projects occur with probability 1 − φ. As is

standard in search models, we define θ ≡ MI/Me. This ratio is important because the

relative availability of each type will determine the probability of deal opportunities and

therefore influence each firms bargaining ability and choice of what type of investor to

become.

Given the availability of investors and entrepreneurs, the number of negotiations to do

a deal each period is given by the matching function ψ(MI ,Me). This function is assumed

to be increasing in both arguments, concave, and homogenous of degree one. This last

assumption ensures that the probability of deal opportunities depends only on the relative

scarcity of the investors to entrepreneurs, θ, which in turn means that the overall size of the

market does not impact investors or entrepreneurs in a different manner. Each individual

investor experiences the same probability of finding an entrepreneur each period, and vice

versa. Thus we define the probability that an investor finds an entrepreneur in any period

as

ψ(MI ,Me)/MI = ψ(1,Me

MI) ≡ qI(θ), (3)

By the properties of the matching function, q′I(θ) ≤ 0, the elasticity of qI(θ) is between

zero and unity, and qI satisfies standard Inada conditions. Thus, an Investor is more

likely to meet an entrepreneur if the ratio of investors to entrepreneurs is low. From an

entrepreneurs point of view the probability of finding an investor is θqI(θ) ≡ qe(θ). This

differs from the viewpoint of investors because of the difference in their relative scarcity.

q′e(θ) ≥ 0, thus entrepreneurs are more likely to meet investors if the ratio of investors to

entrepreneurs is high.

We assume that the measure of each type of investor and project is unchanging. There-

28 JULY 2013

fore, the expected profit from searching is the same at any point in time. Formally, this

stationarity requires the simultaneous creation of more investors to replace those out of

money and more entrepreneurs to replace those who found funding.20 We can think of

these as new funds, new entrepreneurial ideas or old projects returning for more money.21

When an investor and an entrepreneur find each other they must negotiate over any

surplus created and settle on an αS . The surplus created if the investor is committed is,

ξC(p1, VS , VF , E[p2 | S], E[p2 | F ], X, Y, r, uO, uE , uL)

= p1VSE[p2 | S] + (1− p1)VFE[p2 | F ]− 2(uO − uE)− Y (1 + r)−X(1 + r)2. (4)

While the surplus created if the investor is not committed is

ξK(p1, VS , VF , E[p2 | S], E[p2 | F ], X, Y, r, uO, uE , uL)

= p1VSE[p2 | S]− 2(uO − uE) + (1− p1)∆uF − p1Y (1 + r)−X(1 + r)2. (5)

With an abuse of notation we will refer to the surplus created by investments in type

A projects as either ξCA or ξKA depending on whether the investor is committed or a

killer, and the surplus in type B projects as either ξCB or ξKB. The difference between

type A and B projects is ξCA > ξKA while ξKB > ξCB. That is, type A projects generate

more surplus if they receive investment from a committed investor, while type B projects

generate more total surplus if they receive investment from a uncommitted investor.22

20Let mj denote the rate of creation of new type j players (investors or entrepreneurs). Stationarity requires theinflows to equal the outflows. Therefore, mj = qj(θ)Mj .

21In the context of a labor search model, this assumption would be odd, since labor models are focused on therate of unemployment. There is no analog in venture capital investing, since we are not interested in the ‘rate’ thatdeals stay undone.

22This assumption is the first that is unusual in the context of search and matching models. Typically thesemodel assume some form of supermodularity (i.e., ξCA + ξKB > ξKA + ξCB). However, in our model we allowinvestors to choose their type so one type should not be assumed to be superior ex-ante.


The set of possible agreements is Π = (πgf , πfg) : 0 ≤ πgf ≤ ξgf and πfg = ξgf −πgf,

where πgf is the share of the expected surplus of the project earned by the investor and

πfg is the share of the expected surplus of the project earned by the entrepreneur, where

g ∈ [K,C] and f ∈ [A,B].

In equilibrium, if an investor and entrepreneur find each other it is possible to strike a

deal as long as the utility from a deal is greater than the outside opportunity for either. If

an investor or entrepreneur rejects a deal then they return to searching for another partner

which has an expected value of πK , πC , πA, or πB depending on the player.

This matching model will demonstrate the potential for different venture capital industry

outcomes. Although we have reduced the project space down to two projects, this variation

is enough to demonstrate the insight.

To determine how firms share the surplus generated by the project we use the Nash

bargaining solution, which in this case is just the solution to

max(πgf ,πfg)∈Π

(πgf − πg)(πfg − πf ). (6)

The well known solution to the bargaining problem is presented in the following Lemma.23

LEMMA 2: In equilibrium the resulting share of the surplus for an investor of type g ∈

[K,C] investing in a project of type f ∈ [A,B] is

πgf =1

2(ξgf − πf + πg), (7)

while the resulting share of the surplus for the entrepreneur is πfg = ξgf − πgf where the

πg, πf are the disagreement expected values and ξgf is defined by equations (4) and (5).

23The generalized Nash bargaining solution is a simple extension but adds no insight and is omitted.

30 JULY 2013

Given the above assumptions we can write the expected profits both types of investors

and the entrepreneurs, with either type of project, expect to receive if they search for the

other.

πg =qI(θ) [φmax (πgA, πg) + (1− φ) max (πgB, πg)]

1 + r+

1− qI(θ)1 + r

πg (8)

If we postulate that ω fraction of investors choose to be killers, then

πf =qe(θ) [ωmax (πfK , πf ) + (1− ω) max (πfC , πf )]

1 + r+

1− qe(θ)1 + r

πf , (9)

where g ∈ [K,C] and f ∈ [A,B]. These profit functions are also the disagreement utility

of each type during a deal negotiation. With these equations we now have enough to solve

the model.

We present the solution to the matching model in the following proposition.

PROPOSITION 8: If search costs are low relative to the value created by joining B

projects with a killer 2r(1−φ)qI(θ)+ωqe(θ) <

ξKB−ξCBξCB

and A projects with a committed investor

2rφqI(θ)+(1−ω)qe(θ) <

ξCA−ξKAξKA

, then at the equilibrium ω, 1 > ω∗ > 0, there is assortative

matching and committed investors invest in A type projects and killer investors invest in

B type projects. Furthermore, the equilibrium profits of investors who commit (C) or kill

(K) are

πK =(1− φ)qI(θ)

2r + (1− φ)qI(θ) + ωqe(θ)ξKB (10)

πC =φqI(θ)

2r + φqI(θ) + (1− ω)qe(θ)ξCA (11)

And the profits of the entrepreneurs with type A or B projects are

πA =(1− ω)qe(θ)

2r + φqI(θ) + (1− ω)qe(θ)ξCA (12)


πB =ωqe(θ)

2r + (1− φ)qI(θ) + ωqe(θ)ξKB (13)

We leave the proof to the appendix (A.vii). The point of this set up is to have an

equilibrium in which the level of competition from other investors determines the profits

from being a committed or uncommitted investor. In our simple search and matching

model the fraction each investor or entrepreneur gets is endogenously determined by each

players ability to find another investor or investment. The result is an intuitive equilibrium

in which each player gets a fraction of the total surplus created in a deal, ξpf , that depends

on their ability to locate someone else with which to do a deal.

It is interesting to note that even though entrepreneurs would prefer a committed in-

vestor, and investors would prefer to be able to kill a project, since commitment is priced

in equilibrium, there may be a role for both types of investors.

In order for some investors to choose to be killers while others choose to be committed

the expected profits from choosing either type must be the same. If not, investors will

switch from one type to the other, raising profits for one type and lowering them for the

other, until either there are no investors of one type or until the profits equate. Therefore,

the equilibrium ω is the ω = ω∗ such that the profits from either choice are equivalent.

COROLLARY 2: The equilibrium fraction of investors who choose to be killers is 0 ≤

ω∗ ≤ 1 where

ω∗ =(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)ξKB − (2r + (1− φ)qI(θ))φqI(θ)ξCA

qe(θ)(1− φ)qI(θ)ξKB + qe(θ)φqI(θ)ξCA(14)

This corollary elucidates the important insight that there are many parameter realization

that would result in equilibria in which some investors choose endogenously to be killers

while others simultaneously choose to be committed investors. It is not that one choice is

superior. Investors are after profits not innovation and thus prices and levels of competition

32 JULY 2013

adjust so that in many cases it can be equally profitable to be a committed investor

who attracts entrepreneurs, but must require a higher fraction of the company, or an

uncommitted investor who is less desirable to entrepreneurs but who asks for a smaller

fraction of the company. Thus, each type of entrepreneur completes a deal with a different

type of investor.

However, it is also interesting to note that there are some equilibria in which no investor

chooses to be a killer. The following corollary, points out that whether or not it is profitable

to be a killer depends on the level of entrepreneurial aversion to early failure.

COROLLARY 3: The equilibrium fraction of of killer investors, ω∗, is a decreasing func-

tion of the average entrepreneurial aversion to early failure. Furthermore, for high enough

average entrepreneurial aversion to early failure the equilibrium fraction of investors who

choose to be killers, ω∗, may be zero even though there is a positive measure of projects

that create more value with a killer (Me(1− φ) > 0).

The formal proof left to appendix A.viii but the intuition is as follows. As the fear, or

stigma, of early failure increases the surplus created with an uncommitted investor falls.

This lowers the profits to being uncommitted so investors exit and become committed

investors until the profits from either choice are again equivalent. However, there comes a

point where even if an uncommitted investor gets all the surplus from a deal, they would

rather be a committed investor even if all other investors are committed (competitive

forces are not as bad for profits as no commitment). At this point no investor will choose

to be a killer.

Thus, economies with high aversion to early entrepreneurial failure may endogenously

contain no investor willing to fund the type of investments that create more surplus with an

uncommitted investor. Note that this equilibrium can occur even if there are entrepreneurs


looking for funding that create more total surplus if funded by a killer. It may be the

case, however, that with high general aversion to early failure all investors find it more

profitable to form a reputation as committed to attract entrepreneurs and thus look for

projects that create more surplus with a committed investor.

The type of project that wont get funded in this economy are those such that ξK > ξC .

This is true if equation (4) > equation (5), or

VFE[p2 | F ]− Y (1 + r) < ∆uF (15)

As we can see, the projects that wont get funded are those with very low NPV after failure

in the first period. These are the projects for which experimentation mattered greatly.

Note that it is NOT the high risk projects that do not get funded - the probability of

success in the first period p1 does not affect the funding condition. Rather, it is those

projects that are NPV positive before the experiment but are significantly NPV negative

if the early experiment failed. These type of experimental projects cannot receive funding.

This result can help explain the amazing dearth of radical innovations emerging from

countries in Europe and Japan. Many believe that the stigma of failure is much higher

in these cultures, but it would seem that at least some entrepreneurs would be willing to

take the risk. However, what our equilibrium implies is that even those entrepreneurs that

are willing to start very experimental projects may find no investor willing to fund that

level of experimentation. This is a key insight – the institutional funding environment

for innovation is an endogenous equilibrium outcome that may result in places or times

with no investors able to fund radical innovation. In equilibrium, all investors choose to

be committed investors to attract the large mass of entrepreneurs who are willing to pay

for commitment (those with less experimental projects). Thus, when a project arrives

34 JULY 2013

that needs an investor with a sharp guillotine to fund it (or it is NPV negative) there

is no investor able to do it! In this equilibrium, financial intermediaries, such as venture

capitalists, can only fund projects that are less experimental.

Anecdotal evidence is consistent with the evidence from this model. For example, much

of the ‘venture capital’ investing outside the US does not go to truly novel radical in-

novations, but rather to ‘me-too’ projects and firms that although small, already have

customers and products not very different from products sold in many parts of the world,

like a chain of eyeglass stores. Martin Varsavsky, one of Europe’s leading technology entre-

preneurs recently said in an interview with Fortune magazine that “Europeans must accept

that success in the tech startup world comes through trial and error. European [investors]

prefer great plans that don’t fail.”24 Furthermore, European entrepreneurs, and even those

in parts of the United States, complain that they must go to the U.S. or specifically to

Silicon Valley to get their ideas funded. In fact Skype, a huge venture backed success,

was started by European entrepreneurs Niklas Zennstrm and Janus Friis and based in

Luxembourg but received its early funding from US venture capitalists (Bessemer Venture

partners and Draper Fisher Jurvetson). Thus, the problem is two-sided; venture capi-

talists look for less experimental projects to form reputations as failure tolerant because

most entrepreneurs want a more failure tolerant backer.

From a social planners or government perspective, the conclusion from this is to both

attempt to lower the stigma from early entrepreneurial failure and also to increase the

profitability of investing in a set of projects that include some failures. For example,

allowing the enhanced use of losses from early stage investments to offset taxes. Potentially

every dollar of loss could offset two dollars of gain. This would make it more profitable to

be an aggressive investor who actively shut down projects.

24http://tech.fortune.cnn.com/2012/08/14/europe-vc/


V. Extensions and Implications

The tradeoff faced by the investors in our model is one that is more widely applicable.

For example, the manifesto of the venture capital firm called ‘The Founder’s Fund’ out-

lines that they have “never removed a single founder.” The intuition from our model would

suggest that this would clearly attract entrepreneurs and encourage them to start experi-

mental businesses. However, it should simultaneously change the type of entrepreneur the

fund is willing to back. If an investor can’t replace the CEO then it would push them

to back an entrepreneur who has much more entrepreneurial experience, such as a serial

entrepreneur with a proven track record of successes. Such a fund should be less willing

to back a young college student with no prior background if they cannot remove him on

the chance that he turns out not to be good at running the company.

As another example, consider the Ph.D. programs of Chicago and MIT. Both are excel-

lent programs, but Chicago has a reputation for cutting a large portion of the incoming

class after the general exams, while MIT tends to graduate most of the students it admits.

Our model does not tell us which produces ‘better‘ professors on average (competition

might suggest they were equal on average), but our point is that Chicago’s choice should

cause them to take more radical or unconventional students, allowing them to enter and

prove themselves in the program, while MIT will tend to admit students who are more

conventionally strong. Students who were admitted to both but chose MIT over Chicago

likely did so in part because they put a greater cost on the possibility of being cut from

the PhD program after the general exams.

In general our ideas apply to any relationship where in there is the need for exploration,

but also the potential to learn from early experiments. In the subsections below, we

briefly discuss implications the model has for portfolio decisions of investors as well as

36 JULY 2013

their decision to spend money to acquire information.

A. Endogenous Fear of Failure

The idea that the stigma of failure is worse in some parts of the world is generally dis-

cussed as a cultural factor. However, Landier (2002) and Gromb and Scharfstein (2002)

show how the stigma of failure may be endogenous. In Landier’s model, there are multiple

equilibria that stem from the following intuition: A low cost of capital encourages entre-

preneurs to only continue with the best projects. Thus the pool of ‘failed’ entrepreneurs

has a higher average quality since they are discarding bad projects in the hope of getting

better ones, which in turn makes the low cost of capital rational. Gromb and Scharfstein

(2002) have a similar combination of labor market and organizational form in their model

but with differences in the explicit costs and benefits. In this extension we build on this

work and add a similar notion into our model to show it can magnify the likelihood of

extreme equilibria with no financiers willing to fund radial experimentation.

We assume that the average aversion to early failure depends on the number of other

entrepreneurs that are starting highly experimental projects and being quickly killed. If

there are a larger number of entrepreneurs starting and failing then the stigma to having

done so decreases. Specifically we assume that ∆uF = uF (ω)− uE , and ∂uF (ω)/∂ω > 0.

Thus the more investors who choose to be killers, the more surplus is created by an

uncommitted investment. The intuitive effects of this extension are discussed below with

supporting math in appendix A.ix.

A particularly interesting case could result in which this effect pushed investors toward

extremes. As fewer investors chose to be killers, the profits from being a killer investor

could in fact fall (even though there would be less competition) because it led entrepren-

eurs’ stigma of failure to rise sufficiently that it outweighed the higher rents available to


killers from the lower competition for uncommitted investments. Investors would then

have even less incentive to be a killer, so even fewer would choose to do so, which would

lower profits further. This positive feedback could potentially result in an equilibrium

with no killers. However, if a large enough group of investors chose to be killers, then

the profits from being a killer could be high enough to sustain an equilibrium with killers.

Moreover, if adding even more killer investors caused the profits from being a killer to rise

by more than those of committers (who now have less competition) then it could be a Nash

equilibrium for all investors to choose to be a killer. When all (or many) investors are

killers then the competition for projects is stronger but the surplus created is also larger

because the stigma to being killed has fallen. If the increase in surplus is large enough to

offset the increase in competition and the profits from being a committed investor are not

rising by more than the profits from being a killer then it is profitable for all investors to

choose to be a killer.

Thus, what we see in Europe and Japan or even from one city to the next may be an

equilibrium in which there are few or no killer investors so the stigma of failure is quite

high. Thus, the logic from Landier (2002) extends to the financial side of the market.

The financial institutions may be a part of the equilibrium that has little entrepreneurial

failure and in fact make it hard for even those few entrepreneurs willing to overcome the

stigma because they cannot find funding.

B. Portfolios

In this subsection we extend the ideas from above to a simple portfolio problem. We

examine two specific VCs, one of whom who has chosen to be a killer and the other who is

a committed investor. They each have $Z to invest. As in a typical venture capital fund,

we assume all returns from investing must be returned to the investors so that the VCs

38 JULY 2013

only have Z to support their projects.

A committed or uncommitted investor who finds an investment must invest X, however,

they have a different expectations about the need to invest Y . The committed investor

must invest Y if the first stage experiment fails (because the market will not), and they

can choose to invest Y if the first stage succeeds. The uncommitted investor can choose

to invest Y if the first stage succeeds and will not invest if the first stage fails. Therefore,

a committed investor with Z to invest can expect to make at most Z/(X + (1 − p1)Y )

investments and at least Z/(X + Y ) investments. While the uncommitted investor will

make at most Z/X investments and expect to make at least Z/((X + p1Y ) investments.

Therefore, we would expect on average for committed investors to take on a smaller

number of less experimental projects while the uncommitted investors take on a larger

number of more experimental projects. Note that both strategies still expect to be equally

profitable. The committed investors own a larger fraction of fewer projects that are more

likely to succeed but have to invest more in them. While uncommitted investors own a

smaller fraction of more projects and only invest more when it is profitable to do so.

Although the different strategies may be equally profitable there is no question that the

committed investor will complete more projects, and the uncommitted investor will have

completed more radical projects. Note, however that this assumes that the entrepreneur is

willing to receive funding for the more radical innovations from an uncommitted investor.

It may be the case that in equilibrium entrepreneurs with very experimental ideas choose

to stay as wage earners and all investors choose to fund less experimental projects. Thus,

it is not clear ex-ante whether killing or committing will fund more innovation, but condi-

tional on the equilibrium containing both, the killing strategy will fund the more radical

experiments.


C. Spending on Information

One could imagine that spending money in an amount greater than X during the first

stage of the project might increase the information gathered from the experiment. That is,

extra spending might increase VSE[p2 | S]− VFE[p2 | F ]. If so, committed investors gain

nothing by increasing the information learned from the experiment because they cannot

act on it. On the other hand, uncommitted investors gain significantly from increasing

VSE[p2 | S] − VFE[p2 | F ] because there is more to bargain over after success in the

first period and they don’t have to invest after failure. Thus, even if increasing VSE[p2 |

S] − VFE[p2 | F ] simultaneously lowers the probability of success in the first period it is

still potentially beneficial to uncommitted investors. This leads to two insights.

First, it implies that a committed investor will spend less on information gathering to

determine if the project is a good idea. This may be a negative if spending money to

determine whether to go ahead changes a project from negative NPV to positive (see

Roberts and Weitzman (1981)). On the other hand, it may be a good thing if the money

spent by the uncommitted investor is a waste of resources that does not change whether

the project goes forward but just changes the share earned by the investor.

Second, uncommitted investors have an incentive to change the project into one that is

more ‘all or nothing’. The uncommitted investor must reward the entrepreneur for the

expected costs of early termination, but does not pay the ex post costs and thus has an

incentive to increase the probability of early failure (and radical success) after the deal is

done. This may result in an inefficient amount of early termination risk. Furthermore, one

might imagine that entrepreneurs that did not recognize the incentive of the killer VCs to

do this might be surprised to find their VC pushing them to take greater early risk.

40 JULY 2013

VI. Conclusion

While past work has examined optimal amount of failure tolerance at the individual

project level, this idealized planer who adjusts the level of failure tolerance on a project-

by-project basis may not occur in many situations. Our contribution is to instead consider

the ex ante strategic choice of a firm, investor or government aiming to promote innovation

or generate profits. We show that a financial strategy of failure tolerance adopted in the

attempt to promote innovation encourages agents to start projects but simultaneously

reduces the principals willingness to fund experimental projects. Ultimately an increase

in failure tolerance could even reduce total innovation.

We show that in equilibrium, failure tolerance has a price that is increasing with the

level of experimentation in the project. This implies two possible sources of market failure.

First, projects that cannot pay the price cannot be started in equilibrium and must be

funded by the government or else the potential innovation will be lost. These will tend

to be radical innovations, since projects that are less experimental can pay the price to

get a failure tolerant investor. Second, entrepreneurs’ aversion to early failure can affect

the equilibrium. Specifically, if a sufficient number of entrepreneurs face a high stigma

of early failure, an equilibrium can arise where all investors choose to be failure tolerant

and hence only incremental innovations are commercialized in the economy– even if some

entrepreneurs are looking for investors to commercialize radical innovations.


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A. Appendix

i. Proof of Proposition 2

Conditional on a given α1 the investor will invest in the second period as long as

VjαjE[p2 | j]− Y (1 + r) > −c where j ∈ S, F

As noted above, c, is the cost faced by the investor when he stops funding a project andit dies. Thus, the minimum fraction the investor will accept in the second period is

α2j =Y (1 + r)− c

VjE[p2 | j](1− α1)− α1

1− α1.

Thus, an investor will not invest in the second period unless the project is NPV positiveaccounting for the cost of shutdown. This suggests that an investor who already owned afraction of the business, α1, from the first period would be willing to take a lower minimumfraction in the second period than a new investor, and potentially accept even a negativefraction. However, there is a fraction η such that the investor is better off letting an outsideinvestor invest (as long as an outside investor is willing to invest) rather that accept asmaller fraction. If VjE[p2 | j] > Y (1 + r) (which is true for j = S) then an outside

investor would invest for a fraction greater than or equal to Y (1+r)VSE[p2|S] . The fraction η that

makes the investor indifferent between investing or not is the η such that

α1(1− η)VSE[p2 | S]) = (η + α1(1− η))VSE[p2 | S]− Y (1 + r)

The left hand side is what the first period investor expects if a new investor purchases ηin the second period. While the right hand side is the amount the first period investorexpects if he purchases η in the second period. The η that makes this equality hold is

η = Y (1+r)VSE[p2|S] . Note that η does not depend on c because the project continues either

way. Thus, after success, an old investor is better off letting a new investor invest than

accepting a fraction less than Y (1+r)VSE[p2|S] .

25 Thus, the correct minimum fraction that the

investor will accept for an investment of Y in the second period after success in the firstperiod is

α2S =Y (1 + r)

VSE[p2 | S].

However, after failure in the first period then VFE[p2 | F ] < Y (1 + r) and no new investorwill invest. Potentially an old (committed) investor would still invest (to avoid paying c)and the minimum fraction he would accept is

α2F =Y (1 + r)− c

VFE[p2 | F ](1− α1)− α1

1− α1.

The entrepreneur, on the other hand, will continue with the business in the second

25This assumes perfect capital markets that would allow a ‘switching’ of investors if entrepreneurs tried to extracttoo much. No results depend on this assumption but it makes the math easier and more intuitive, and we don’twant to drive any results off of financial market frictions.

48 JULY 2013

period as long as,

Vj(1− αj)E[p2 | j] + uE > uF where j ∈ S, F.

Since αj = α2j + α1(1− α2j), for a given α1 the maximum fraction the entrepreneur willgive to the investor in the second period is

α2j = 1− uF − uEVjE[p2 | j](1− α1)

∀ j ∈ S, F.

Similarly to the investor, after success in the first period, there is a point at which theentrepreneur who already owns a fraction 1 − α1 should quit and let the investors hirea new manager rather than take a smaller fraction. Thus, there is a η that makes theentrepreneur indifferent between staying and leaving:

(1− α1)ηVSE[p2 | S] + uF = ((1− η) + (1− α1)η)VSE[p2 | S] + uE

Thus, the correct maximum fraction the entrepreneur will give up in the second periodafter success in the first period is26


However, after failure in the first period the maximum that the entrepreneur is willing togive up to keep the business alive is

α2F = 1− uF − uEVFE[p2 | F ](1− α1)

The entrepreneur cannot credibly threaten to leave after failure unless he must give upmore than α2F , as his departure will just cause the business to be shut down.

ii. Proof of Propositions 1 and 3

Bargaining will result in a fraction in the second period of α2j = γα2j + (1− γ)α2j . Forexample, if the entrepreneur has all the bargaining power, γ = 1, then the investor mustaccept his minimum fraction, α2j = α2j , while if the investor has all the bargaining power,γ = 0, then the entrepreneur must give up the maximum, α2j = α2j . While if each hassome bargaining power then they share the surplus created by the opportunity.

Given this, we can substitute into αj = α2j +α1(1−α2j) and solve for the final fractionsthe investor and entrepreneur will obtain depending on success or failure at the first stage.Substituting we find αj = γα2j + (1 − γ)α2j + α1(1 − (γα2j + (1 − γ)α2j)). This can be

rewritten as αj = [γα2j + (1−γ)α2j ](1−α1) +α1. Substituting in for α2j and α2j we findthat

αS =

[γY (1 + r)

VSE[p2 | S]+ (1− γ)

[1− uF − uE

VSE[p2 | S]

]](1− α1) + α1 (A-1)

26This requires the assumption of perfect labor markets that would allow a ‘switching’ of CEOs among entre-preneurial firms if investors tried to extract too much. No results depend on this assumption but it makes the matheasier and more intuitive, and we don’t want to drive any results off of labor market frictions.


and αF reduces to

αF = γ

[Y (1 + r)− cVFE[p2 | F ]

]+ (1− γ)

[1− uF − uE

VFE[p2 | F ]

](A-2)

Of course, in both cases negotiations must result in a fraction between zero and one.27

Note that αF does not depend on the negotiations in the first period because after failure,renegotiation determines the final fractions.28 Of course, investors and entrepreneurs willaccount for this in the first period when they decide whether or not to participate.29 Wesolve for the first period fractions in appendix A.iii but these are not necessary for theproof.

The solution αF is only correct assuming a deal can be reached between the investorand the entrepreneur in the second period (otherwise the company is shut down afterearly failure). Interesting outcomes will emerge both when an agreement can and cannotbe reached as this will affect both the price of, and the willingness to begin, a project.

Stepping back to the first period, an investor will invest as long as

p1[VSαSE[p2 | S]− Y (1 + r)]−X(1 + r)2

+ (1− p1)[VFαFE[p2 | F ]− Y (1 + r)] ≥ 0 (A-3)

if the 2nd period agreement conditions are met after failure. Or,

p1[VSαSE[p2 | S]− Y (1 + r)]−X(1 + r)2 − (1− p1)c ≥ 0 (A-4)

if they are not.

The entrepreneur will choose to innovate and start the project if

p1[VS(1− αS)E[p2 | S] + uE ] + uE

+ (1− p1)[VF (1− αF )E[p2 | F ] + uE ] ≥ 2uO (A-5)

if the 2nd period agreement conditions are met after failure. Or,

p1[VS(1− αS)E[p2 | S] + uE ] + uE + (1− p1)uF ≥ 2uO (A-6)

if they are not.

The four above equations can be used to solve for the minimum fractions needed bythe investor and entrepreneur both when a deal after failure can be reached and whenit cannot. If the agreement conditions in the 2nd period after failure are met, then theminimum fraction the investor is willing to receive in the successful state and still chooseto invest in the project is found by solving equation (A-3) for the minimum αS such that

27Since negotiations must result in a fraction between zero and one, then if a deal can be done then if γ <(uF−uE)/(Y (1+r)−c−VFE[p2 | F ]+uF−uE) then αF = 1, or if γ < −(uF−uE)/(Y (1+r)−VSE[p2 | S]+uF−uE)then αS = 1. Since c ≤ Y (1 + r) the negotiations will never result in a fraction less than zero.

28In actual venture capital deals so called ‘down rounds’ that occur after poor outcomes often result in a completerearrangement of ownership fractions between the first round, second round and entrepreneur.

29Alternatively we could assume that investors and entrepreneurs predetermine a split for for every first stageoutcome. This would require complete contracts and verifiable states so seems less realistic but would not changethe intuition or implications of our results.

50 JULY 2013

the inequality holds:

αSA=Y (1 + r) +X(1 + r)2 − (1− p1)VFαFE[p2 | F ]

p1VSE[p2 | S]

where the subscript A signifies that an agreement can be reached after first period failure.

The maximum fraction the entrepreneur can give up in the successful state and still bewilling to choose the entrepreneurial project is found by solving equation (A-5) for themaximum αS such that the inequality holds:

αSA= 1− 2(uO − uE)− (1− p1)E[p2 | F ]VF (1− αF )

p1VSE[p2 | S]

where αF is defined in equation (A-2) in both αSAand αSA

. Both αSAand αSA

dependon the negotiations in the failed state, αF , because the minimum share the players needto receive in the the good state to make them willing to choose the project depends onhow badly they do in the bad state.

If a second period agreement after failure cannot be reached then the minimum frac-tion of the investor and the maximum fraction of the entrepreneur are found by solvingequations (A-4) and (A-6) respectively, to find

αSN=p1Y (1 + r) +X(1 + r)2

p1VSE[p2 | S]

and

αSN= 1− (1 + p1)(uO − uE) + (1− p1)(uO − uF )

p1VSE[p2 | S]

where the N subscript represents the fact that no agreement can be reached after failure.

iii. Derivation of first period fractions

The maximum and minimum required shares after first period success, αSi and αSi ,

directly imply first period minimum an maximum fractions, α1i and α1i (i ∈ [A,N ]),

because we already know from above, equation (A-1), that

αS =

[γY (1 + r)

VSE[p2 | S]+ (1− γ)(1− uF − uE

VSE[p2 | S])

](1− α1) + α1

Thus, we can solve for the α1 that just gives the investor his minimum αS . Let Z equalthe term in brackets in the equation above and we can solve for α1 as a function of αS .

α1 =αS − Z1− Z

(A-7)

Plugging in αSAfor αS yields the minimum required investor fraction α1A :

α1A =

Y (1+r)+X(1+r)2−(1−p1)VFαFE[p2|F ]p1VSE[p2|S] − Z

1− Z


as a function of αF . And substituting in for αF from equation (A-2) and Z from aboveyields,

α1A = 1− p1VSE[p2 | S]− p1Y (1 + r)−X(1 + r)2 − (1− p1)γc

p1(γVSE[p2 | S]− γY (1 + r) + (1− γ)(uF − uE))

− (1− p1)(1− γ)(VFE[p2 | F ]− Y (1 + r)− (uF − uE))


This is the minimum fraction required by the investor assuming that a deal can be achievedin the second period after failure in the first period.30 In equilibrium the investor’s min-imum depends on the entrepreneur’s gains and costs because they must negotiate andparticipate.

If instead, an agreement cannot be reached after failure in the first period then theproject is stopped. In this case the minimum fraction required by the investor can befound by plugging αSN

into equation (A-7) for αS , where αSNis the minimum when no

second period deal can be reached. In this case the minimum required investor fractionα1N is

α1N =

p1Y (1+r)+X(1+r)2

p1VSE[p2|S] − Z1− Z

or,

α1N = 1− p1VSE[p2 | S]− p1Y (1 + r)−X(1 + r)2


We can similarly calculate the maximum fraction the entrepreneur is willing to give upin the first period. The maximum fraction can be found by plugging αSi into equation (A-7) for αSi , where αSi (i ∈ [A,N ]) is the maximum when either a second period agreementafter failure can (A) or cannot (N) be reached. When a second period agreement can bereached α1A is

α1A = 1− 2(uO − uE)− (1− p1)E[p2 | F ]VF (1− αF )


And when a second period deal after failure cannot be reached α1N is

α1N = 1− (1 + p1)(uO − uE) + (1− p1)(uO − uF )


30Technical note: with extreme values it is possible that αF would be greater than 1 or less than zero. In thesecases αF is bound by either zero or 1. This would cause the α1 to increase or decrease. This dampens some ofthe effects in extreme cases but alters no results. To simplify the exposition we assume that parameters are in thereasonable range such that the investor and entrepreneur would not be willing to agree to a share greater than 1 orless than zero.

52 JULY 2013

iv. Proof of Proposition 5:

For a moment assume that the uncommitted investor has an explicit cost of early ter-mination of c (where c is zero). In which case

αSA− αSN

=(1− p1)(Y (1 + r)− VFαFE[p2 | F ]− c)

p1VSE[p2 | S]> 0

for any 0 ≤ αF ≤ 1. If we increase c to c to make the investor failure tolerant, then justat the point where VFE[p2 | F ] − Y (1 + r) + c = 0 it is still the case that αSA

≥ αSN,

because αF ≤ 1. Therefore, since ∂αSN/∂c > 0, αSN

(c) > αSN(c). Therefore, since

∂αSA/∂c > 0∀c we know that αSA

(c ≥ c) > αSN(c) > αSN

(c). Thus, the minimum acommitted investor is willing to accept is always greater than the minimum of an investorwho is uncommitted i.e., αSA

(c ≥ c) > αSN(c). And equation (A-7) demonstrates that a

smaller αSi results in a smaller α1. Furthermore, Lemma 1 implies that an uncommittedinvestor cannot do a deal after early failure. QED

v. Proof of Proposition 6:

It is clearly possible that both αSA− αSA

< 0 and αSN− αSN

< 0. For example, a

project with a low enough VS and/or VF (or high X) could have both differences less thanzero for any positive c (i.e., independent of the failure tolerance of the investor). Similarly,for a high enough VS and/or VF (or low X) both αSA

− αSA> 0 and αSN

− αSN> 0,

even for c equal to the maximum c of Y (1 + r). Thus, extremely bad projects will not bestarted and extremely good projects will be started by any type of investor.

Committed investors, who will reach an agreement after early failure, will start theproject if αSA

− αSA≥ 0. Uncommitted investors, who will kill the project after early

failure, will start the project if αSN− αSN

≥ 0. The difference between αSA− αSA

andαSN

− αSNis

(1− p1)VFE[p2 | F ]− (1− p1)∆uF − (1− p1)Y (1 + r)

p1VSE[p2 | S](A-8)

For an uncommitted investor, equation (A-8) may be positive or negative depending onthe relative magnitudes of VFE[p2 | F ], ∆uF , and Y (1 + r). If it is positive, then for someparameters αSA

− αSA≥ 0 while αSN

− αSN< 0. In these cases the project can only be

funded by a committed investor. If the difference in equation (A-8) is negative then forsome parameters αSA

−αSA< 0 while αSN

−αSN≥ 0. In these cases the project can only

be funded by an uncommitted investor. QED

vi. Proof of Proposition 7:

A project can be funded by a committed investor if αSA− αSA

≥ 0. For two projectswith the same expected payout, αSA

− αSAhas the same sign, i.e., both projects either

can or cannot be funded and changing VSE[p2 | S] − VFE[p2 | F ] does not change that.This can be seen by noting that the numerator of equation (1) is unaffected by changes inVSE[p2 | S]−VFE[p2 | F ] as long as p1VSE[p2 | S] + (1− p1)VFE[p2 | F ] does not change.


A project can be funded by an uncommitted investor if αSN−αSN

≥ 0. If VSE[p2 | S]−VFE[p2 | F ] increases but p1, and the expected payout, p1VSE[p2 | S]+(1−p1)VFE[p2 | F ],stay the same, then VSE[p2 | S] must have increased and VFE[p2 | F ] must have decreased.In which case αSN

−αSNincreased (see equation (2)) and the difference between αSA

−αSA

and αSN−αSN

(equation (A-8)) decreased. Therefore, there are a larger set of parameterssuch that αSA

− αSA< 0 while αSN

− αSN≥ 0, i.e., the project can only be funded by an

uncommitted investor. QED

vii. Proof of Proposition 8:

Let g ∈ [K,C] represent the investor type investing in a project of type f ∈ [A,B],and let ω represent the fraction of investors that choose to be killers (K) rather thancommitted (C). Also, let πg and πf represent the expected profits of the investor andentrepreneur respectively before they find a partner, while πgf and πfg represent theirrespective expected profits conditional on doing a deal with a partner of type f or grespectively.

We begin with equation (8) and assume that πC < πCA, πC > πCB, πK < πKB,πK > πKA, which we later verify in equilibrium. Thus,

πK =qI(θ) [φπK + (1− φ)πKB]

1 + r+

1− qI(θ)1 + r

πK (A-9)

πC =qI(θ) [φπCA + (1− φ)πC ]

1 + r+

1− qI(θ)1 + r

πC (A-10)

Next we use equation (9) and assume that πA < πAC , πA > πBC , πB < πBK , πB > πBC ,which we also verify in equilibrium.

πA =qe(θ) [ωπA + (1− ω)πAC ]

1 + r+

1− qe(θ)1 + r

πA (A-11)

πB =qe(θ) [ωπBK + (1− ω)πB]

1 + r+

1− qe(θ)1 + r

πB (A-12)

Using Lemma 2 and solving we find


2r + (1− φ)qI(θ)[ξKB − πB] (A-13)

πC =φqI(θ)

2r + φqI(θ)[ξCA − πA] (A-14)


2r + (1− ω)qe(θ)[ξCA − πC ] (A-15)

πB =ωqe(θ)

2r + ωqe(θ)[ξKB − πK ] (A-16)

Therefore, with 4 equations and 4 unknowns we can solve for πK , πC , πA, and πB.


2r + (1− φ)qI(θ) + ωqe(θ)ξKB (A-17)

54 JULY 2013

πC =φqI(θ)

2r + φqI(θ) + (1− ω)qe(θ)ξCA (A-18)


2r + φqI(θ) + (1− ω)qe(θ)ξCA (A-19)

πB =ωqe(θ)

2r + (1− φ)qI(θ) + ωqe(θ)ξKB (A-20)

These are the equilibrium profits, but we must confirm that πCB < πC < πCA andπKA < πK < πKB as well as πAK < πA < πAC and πBC < πB < πBK .

Lemma 2 tells us that πgf = 12(ξgf − πf + πg) and πfg = ξgf − πgf . Thus checking all

the inequalities just above reduces to checking that ξCB − πB < πC < ξCA − πA and thatξKA − πA < πK < ξKB − πB. Substituting for πK , πC , πA, and πB from above we seethat it is always the case that πC < ξCA − πA and πK < ξKB − πB as long as ξKB andξCA are positive (i.e. a deal creates value). Furthermore, since at the equilibrium ω = ω∗

it must be the cast that πC = πK , therefore, ξCB − πB < πC as long as

2r

(1− φ)qI(θ) + ωqe(θ)<ξKB − ξCB

ξCB(A-21)

and ξKA − πA < πK as long as

2r

φqI(θ) + (1− ω)qe(θ)<ξCA − ξKA

ξKA(A-22)

viii. Proof of Corollary 3:

Since ∂πK/∂ω < 0 and ∂πC/∂ω > 0, single crossing is insured and ω∗ is determined thepoint at which πK = πC using the results from proposition 8. As corollary 2 notes thisresults in

ω∗ =(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)ξKB − (2r + (1− φ)qI(θ))φqI(θ)ξCA

qe(θ)(1− φ)qI(θ)ξKB + qe(θ)φqI(θ)ξCA(A-23)

Or, ω∗ = 1 if πK(1) > πC(1) or ω∗ = 0 if πK(0) > πC(0).

As the fear or stigma of early failure decreases, ∆uF increases. Using equations (4) and

(5) we find that∂ξCf

∂∆uF= 0 and

∂ξKf

∂∆uF= (1− p1) > 0. Therefor,

∂ω∗

∂∆uF=

(qe(θ)(1− φ)qI(θ)ξKB)(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)∂ξKB∂∆uF

(qe(θ)(1− φ)qI(θ)ξKB + qe(θ)φqI(θ)ξCA)2

+(qe(θ)φqI(θ)ξCA)(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)

∂ξKB∂∆uF


−[(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)ξKB]qe(θ)(1− φ)qI(θ)

∂ξKB∂∆uF


+[(2r + (1− φ)qI(θ))φqI(θ)ξCA]qe(θ)(1− φ)qI(θ)

∂ξKB∂∆uF

(qe(θ)(1− φ)qI(θ)ξKB + qe(θ)φqI(θ)ξCA)2(A-24)


And since the first and third terms are the same but with opposite sign this reduces to

∂ω∗

∂∆uF=

(qe(θ)φqI(θ)ξCA)(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)∂ξKB∂∆uF


+[(2r + (1− φ)qI(θ))φqI(θ)ξCA]qe(θ)(1− φ)qI(θ)

∂ξKB∂∆uF

(qe(θ)(1− φ)qI(θ)ξKB + qe(θ)φqI(θ)ξCA)2(A-25)

which is positive since both terms are positive. The proves the first part of the corollary.No investor chooses to be a killer if

(2r + φqI(θ) + qe(θ))(1− φ)qI(θ)ξKB ≤ (2r + (1− φ)qI(θ))φqI(θ)ξCA (A-26)

Since the above solved for ∂ω∗

∂∆uF> 0 it is easy to see that ∆uF is not in the numerator of

∂ω∗

∂∆uFand therefore the second derivative of ω∗ with respect to ∆uF is negative everywhere.

Therefore, for small enough ∆uF condition (A-26) holds and no investor chooses to be akiller. Note that condition (A-26) can hold even though ξKB > 0 and even though φ < 1so there are projects that create more surplus with a killer as an investor.

ix. Support for extension V.A:

The extension assumes that ∆uF = uF (ω) − uE , and ∂uF (ω)/∂ω > 0. Therefore, thesign of ∂πK/∂ω is no longer deterministic because ξKB now depends on ω and thus theprofits from being a killer are affected in two ways by an increase in ω.


2r + (1− φ)qI(θ) + ωqe(θ)ξKB(ω) (A-27)

In the numerator ξKB(ω) increases with ω (i.e., total surplus increases) but the denomi-nator also increase (i.e. competition increases).

However, ∂πC/∂ω > 0. That is, the profits from being committed always increase ifmore investors choose to be killers. This is because

πC =φqI(θ)

2r + φqI(θ) + (1− ω)qe(θ)ξCA (A-28)

only has ω in the denominator (with a negative sign) because ξCA is not effected by ω(see equation (4)).

Therefore, the Nash equilibria, which we will refer to as the set Ω∗, includes all the ω∗

such that πK(ω∗) = πC(ω∗) and ∂πK/∂ω∗ ≤ ∂πC/∂ω∗. That is, as long as increasing the

fraction of killers increases killer profits, but committer profits increase by the same ormore, then no investor will want to change on the margin.

Furthermore, the end points (ω∗ = 1 or ω∗ = 0) if πK(1) > πC(1) or πK(0) > πC(0).Note first that it is possible that both πK(1) > πC(1) or πK(0) > πC(0) and thus

that both ω∗ = 1 and ω∗ = 0 are equilibria. This would occur for example if the profitsfrom being a killer are less than a committer, if all investors are committers, but killerprofits always increase as more investors become killers ∂πK/∂ω > 0 until it is above theprofit from being committed when all investors are killers. In this case both extremes are

56 JULY 2013

equilibria and no other point is an equilibria.It is also possible that there are many points that are Nash equilibria. All that is required

is an ω such that a change in the fraction of investors on the margin is not beneficial tothe changing investor. We can find all points πK(ω∗) = πC(ω∗)

(1− φ)qI(θ)

2r + (1− φ)qI(θ) + ωqe(θ)ξKB(ω) =

φqI(θ)

2r + φqI(θ) + (1− ω)qe(θ)ξCA (A-29)

and

− (1− φ)qI(θ)qe(θ)

(2r + (1− φ)qI(θ) + ωqe(θ))2ξKB(ω) +

(1− φ)qI(θ)

2r + (1− φ)qI(θ) + ωqe(θ)

∂ξKB(ω)

∂ω

≤ φqI(θ)qe(θ)

(2r + φqI(θ) + (1− ω)qe(θ))2ξCA (A-30)

Remember that

ξKB = p1VSE[p2 | S]− 2(uO − uE) + (1− p1)∆uF − p1Y (1 + r)−X(1 + r)2. (A-31)

So∂ξKB∂ω

= (1− p1)∂uF (ω)

∂ω. (A-32)

Thus,

− (1− φ)qI(θ)qe(θ)

(2r + (1− φ)qI(θ) + ωqe(θ))2ξKB(ω) +

(1− φ)qI(θ)

2r + (1− φ)qI(θ) + ωqe(θ)(1− p1)

∂uF (ω)

∂ω

≤ φqI(θ)qe(θ)

(2r + φqI(θ) + (1− ω)qe(θ))2ξCA (A-33)

And we can see that the inequality is true when ∂uF (ω)∂ω is small, but may not hold at all

points where πK(ω∗) = πC(ω∗). Essentially the only equilibria are points where πK(ω∗)crosses πC(ω∗) from above or is tangent from below.

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